For a < 1, the curve is the oval of Booth.
For a > 1, the curve is the lemniscate of
Booth.
When the parameter a has the value 2, the curve is the lemniscate
of Bernoulli.
Proclus (75 BC) and Eudoxus were the first to investigate the curve.
After Proclus the curve has been named hippopede of Proclus.

In Cartesian coordinates the curve has the form: (x2 + y2)2
= b x2 + y2 with b = 1 - a.
And we see that the curve is a bicircular
(rational) quartic.